| L(s) = 1 | − 2·4-s + 4.35·7-s − 3·9-s + 5·11-s + 4·16-s + 4.35·17-s + 8.71·23-s − 8.71·28-s + 6·36-s + 13.0·43-s − 10·44-s + 4.35·47-s + 12.0·49-s − 15·61-s − 13.0·63-s − 8·64-s − 8.71·68-s + 13.0·73-s + 21.7·77-s + 9·81-s − 8.71·83-s − 17.4·92-s − 15·99-s − 10·101-s + 17.4·112-s + 19.0·119-s + ⋯ |
| L(s) = 1 | − 4-s + 1.64·7-s − 9-s + 1.50·11-s + 16-s + 1.05·17-s + 1.81·23-s − 1.64·28-s + 36-s + 1.99·43-s − 1.50·44-s + 0.635·47-s + 1.71·49-s − 1.92·61-s − 1.64·63-s − 64-s − 1.05·68-s + 1.53·73-s + 2.48·77-s + 81-s − 0.956·83-s − 1.81·92-s − 1.50·99-s − 0.995·101-s + 1.64·112-s + 1.74·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.420480523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.420480523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13.0T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84163220166295224049137351703, −7.26219499095555077411811779446, −6.19312827325096885217432854483, −5.50653509629120021647298524263, −4.98090486488464700170258968115, −4.30400428808585718401726197552, −3.61122366219817124898056168815, −2.70479366693480888803891032607, −1.41705652399818893920402706597, −0.883885917930079300455782817475,
0.883885917930079300455782817475, 1.41705652399818893920402706597, 2.70479366693480888803891032607, 3.61122366219817124898056168815, 4.30400428808585718401726197552, 4.98090486488464700170258968115, 5.50653509629120021647298524263, 6.19312827325096885217432854483, 7.26219499095555077411811779446, 7.84163220166295224049137351703
